given by and we choose as the other leg then the hypotenuse is . This is essentially the generation formula above with set to 1 and allowing to range from 2 to infinity. Pythagorean triangles with integer altitude from the hypotenuse There are no primitive Pythagorean triangles with integer altitude from the hypotenuse. This is because twice the area equals any base times the corresponding height: 2 times the area thus equals both ab and cd where d is the height from the hypotenuse c. The three side lengths of a primitive triangle are coprime, so d = is in fully reduced form; since c cannot equal 1 for any primitive Pythagorean triangle, d cannot be an integer. However, any Pythagorean triangle with legs x, y and hypotenuse z can generate a Pythagorean triangle with an integer altitude, by scaling up the sides by the length of the hypotenuse z. If d is the altitude, then the generated Pythagorean triangle with integer altitude is given by Consequently, all Pythagorean triangles with legs a and b, hypotenuse c, and integer altitude d from the hypotenuse, with gcd(a, b, c, d) = 1, which necessarily satisfy both a2 + b2 = c2 and , are generated by for coprime integers m, n with m > n. Heronian triangles with sides in arithmetic progression A triangle with integer sides and integer area has sides in arithmetic progression if and only if the sides are (b – d, b, b + d), where and where g is the greatest common divisor of and Heronian triangles with one angle equal to twice another All Heronian triangles with B = 2A are generated by either with integers k, s, r such that s2 > 3r2, or , , , , with integers such that and No Heronian triangles with B = 2A are isosceles or right triangles because all resulting angle combinations generate angles with non-rational sines, giving a non-rational area or side. Isosceles Heronian triangles All isosceles Heronian triangles are decomposable. They are formed by joining two congruent Pythagorean triangles along either of their common legs such that the equal sides of the isosceles triangle are the hypotenuses of the Pythagorean triangles, and the base of the isosceles triangle is twice the other Pythagorean leg. Consequently, every Pythagorean triangle is the building block for two isosceles Heronian triangles since the join can be along either